 Croot-Sisask almost-periodicity theorem

If finite sets $A,S$ in a group satisfy $|A+S|\le K|A|$, then for $q\ge 2$ and $0<ϵ<1$ there is $T\subseteq S$ with $|T|\ge (2K{)}^{-O(q/{ϵ}^2)}|S|$ such that every $t\in T-T$ is an $L^q$ almost period of $1_A\ast f$: